Theorem 2initoinv 17957

Description: Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)

Hypotheses

Ref	Expression
initoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoeu1.a	⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
initoeu1.b	⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Assertion

Ref	Expression
2initoinv	⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Proof of Theorem 2initoinv

Step	Hyp	Ref	Expression
1		eqid 2733	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2733	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2733	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		initoeu1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
6		initoeu1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
7		initoo 17954	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
8	4, 6, 7	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
9	8	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
10		initoeu1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))
11		initoo 17954	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	4, 10, 11	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	12	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
14		simp3 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵))
15		simp2 1138	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴))
16	1, 2, 3, 5, 9, 13, 9, 14, 15	catcocl 17626	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴))
17	1, 2, 4	initoid 17948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (InitO‘𝐶)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
18	6, 17	mpdan 686	. . . . . . 7 ⊢ (𝜑 → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
19	18	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
20	19	eleq2d 2820	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴) ↔ (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)}))
21		elsni 4645	. . . . 5 ⊢ ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)} → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
22	20, 21	syl6bi 253	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴)))
23	16, 22	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
24		eqid 2733	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
25		eqid 2733	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
26	1, 2, 3, 24, 25, 5, 9, 13, 14, 15	issect2 17698	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ↔ (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴)))
27	23, 26	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Sect‘𝐶)𝐵)𝐺)
28	1, 2, 3, 5, 13, 9, 13, 15, 14	catcocl 17626	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵))
29	1, 2, 4	initoid 17948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (InitO‘𝐶)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
30	10, 29	mpdan 686	. . . . . . 7 ⊢ (𝜑 → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
31	30	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
32	31	eleq2d 2820	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵) ↔ (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)}))
33		elsni 4645	. . . . 5 ⊢ ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)} → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
34	32, 33	syl6bi 253	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵)))
35	28, 34	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
36	1, 2, 3, 24, 25, 5, 13, 9, 15, 14	issect2 17698	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(𝐵(Sect‘𝐶)𝐴)𝐹 ↔ (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵)))
37	35, 36	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)
38		eqid 2733	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
39	1, 38, 4, 8, 12, 25	isinv 17704	. . 3 ⊢ (𝜑 → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
40	39	3ad2ant1 1134	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
41	27, 37, 40	mpbir2and 712	1 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {csn 4628  ⟨cop 4634   class class class wbr 5148  ‘cfv 6541  (class class class)co 7406  Basecbs 17141  Hom chom 17205  compcco 17206  Catccat 17605  Idccid 17606  Sectcsect 17688  Invcinv 17689  InitOcinito 17928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-cat 17609  df-cid 17610  df-sect 17691  df-inv 17692  df-inito 17931

This theorem is referenced by:  initoeu1  17958